On the Inverse Resonance Problem for Schrödinger Operators

Abstract

We consider Schrödinger operators on [0, ∞) with compactly supported, possibly complex-valued potentials in L 1 ([0, ∞)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a byproduct we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.

Resonances for Dirac operators on the half-line

“…Inverse problems (characterization, recovering, plus uniqueness) in terms of resonances were solved by Korotyaev for the Schrödinger operator with a compactly supported potential on the real line [K05] and the half-line [K04]. The "local resonance" stability problems were considered in [K04s], [MSW10].…”

On the Solvability Complexity Index, the 𝑛-pseudospectrum and approximations of spectra of operators

“…As the importance of non-Hermitian operators in physics has been established, the spectral theory of such operators has been given a substantial amount of attention [21,23,24,16]. (Note also that non-Hermitian spectral problems in quantum mechanics occur in the theory of resonances [52,40]). Since the spectral theory of non-Hermitian operators is very different from the selfadjoint case, very little is known in general, and the same is true for the theory of approximating spectra.…”

Inverse resonance problems for the Schrödinger operator on the real line with mixed given data

“…Inverse resonance problem for the Schrödinger operator consists in determining the potential q from the eigenvalues and resonances and/or other observable data, which is an important part of inverse scattering theory [2,3,11]. Let's mention that inverse resonance problem for the Schrödinger operator on the half line has been studied (see, for example, [7,8,12,14,15,17] and the references therein). In the half line case, the unique recovery of the potential from the eigenvalues and resonances is valid [7,14].…”